计算机问题龙解一论题3-18 群同态基本定理 2017年3月13日
计算机问题求解 – 论题3-18 - 群同态基本定理 2017年3月13日
问题1：我们为什么定义这个函数是“同构”？ iso-morphology Two groups (G,)and (H,o)are isomorphic if there exists a one-to-one and onto map GH such that the group operation is preserved;that is, (a·b)=(a)o(b) 同构其实可在 for all a and b in G.If G is isomorphic to H is called an isomorphism. 任何代数结构 (系统)上讨论
问题1：我们为什么定义这个函数是“同构” ？ iso-morphology 同构其实可在 任何代数结构 (系统)上讨论
问题2：从这个定理中，你能解释我们为 什么研究“同构”吗？ Theorem 9.1 Let G-H be an isomorphism of two groups.Then the following statements are true. 1.-1:HG is an isomorphism. 2.G=H. 3.If G is abelian,then H is abelian. 4.If G is cyclic,then H is cyclic. 5.If G has a subgroup of order n,then H has a subgroup of order n
问题2：从这个定理中，你能解释我们为 什么研究“同构”吗？
如何判断两个系统的同构？ Theorem 9.2 All cyclic groups of infinite order are isomorphic to Z. PRoOF.Let G be a cyclic group with infinite order and suppose that a is a generator of G.Define a map:ZG byo:n a".Then 观察 o(m+n)amtn a"a"o(m)o(n). 构造 To show that o is injective,suppose that m and n are two elements in Z, where m n.We can assume that m >n.We must show that am a". 证明 Let us suppose the contrary;that is,am=a".In this case am-=e,where m-n >0,which contradicts the fact that a has infinite order.Our map is onto since any element in G can be written as an for some integer n and o(n)a". ▣
观察 构造 证明 如何判断两个系统的同构？
问题3.1：这个定理给我们什么感觉？ Theorem 9.5 The isomorphism of groups determines an equivalence rela- tion on the class of all groups. 如何去证明这个定理？
问题3.1：这个定理给我们什么感觉？ 如何去证明这个定理？